Conic Section Equations

Why This Matters

Conic sections aren't just abstract curves—they're the mathematical foundation for everything from satellite dish design to planetary orbits to headlight reflectors. In Honors Algebra II, you're being tested on your ability to recognize these curves from their equations, convert between forms, and understand how changing parameters affects the shape. The core concepts here—distance relationships, symmetry, and the interplay between algebraic and geometric representations—will follow you straight into precalculus and calculus.

Don't just memorize formulas. For each conic, know what geometric property defines it (equidistant points? constant sum of distances?) and how to identify it from an equation. When you see a problem, ask yourself: What's the center? What's the orientation? What do the constants tell me about the shape? Master these questions, and you'll handle any conic section problem thrown at you.

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Curves Defined by Equal Distance

These conics are built on the simplest geometric idea: points that maintain a specific distance relationship to a center or axis.

Circle Equation

• Standard form: $(x - h)^2 + (y - k)^2 = r^2$—every point is exactly $r$ units from the center $(h, k)$
• Radius $r$ determines size; the equation uses $r^2$, so don't forget to take the square root when finding the actual radius
• Only conic with equal coefficients—when $A = C$ and $B = 0$ in general form, you've got a circle

Parabola Equation (Vertical Axis)

• Standard form: $(x - h)^2 = 4p(y - k)$—the set of points equidistant from the focus and directrix
• Parameter $p$ is the distance from vertex to focus; positive $p$ opens upward, negative opens downward
• Vertex at $(h, k)$ with focus at $(h, k + p)$ and directrix at $y = k - p$

Parabola Equation (Horizontal Axis)

• Standard form: $(y - k)^2 = 4p(x - h)$—same distance property, but oriented left-right
• Sign of $p$ determines direction—positive opens right, negative opens left
• Focus at $(h + p, k)$ with directrix at $x = h - p$; the squared variable tells you the axis of symmetry

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Curves Defined by Sum or Difference of Distances

Ellipses and hyperbolas are defined by their relationship to two foci—the key is whether you're adding or subtracting those distances.

Ellipse Equation

• Standard form: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$—sum of distances to both foci is constant (equals $2a$)
• Larger denominator determines orientation—if $a^2 > b^2$, major axis is horizontal; swap the inequality, it's vertical
• Relationship $c^2 = a^2 - b^2$ gives the focal distance; $a$ is always the semi-major axis

Hyperbola Equation (Horizontal Transverse Axis)

• Standard form: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$—difference of distances to foci is constant (equals $2a$)
• Positive term determines orientation—$x$ term positive means branches open left and right
• Asymptotes: $y - k = \pm\frac{b}{a}(x - h)$—these lines guide the hyperbola's shape at infinity

Hyperbola Equation (Vertical Transverse Axis)

• Standard form: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$—same difference property, branches open up and down
• Relationship $c^2 = a^2 + b^2$ for hyperbolas; note this is addition, unlike ellipses
• Asymptotes: $y - k = \pm\frac{a}{b}(x - h)$—slope formula flips compared to horizontal hyperbolas

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Classification and Conversion Tools

These formulas help you identify conic types and convert between equation forms—essential skills for exam problems.

General Form of a Conic Section

• Form: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$—the universal equation that contains all conics
• Discriminant $B^2 - 4AC$ classifies the conic—negative = ellipse/circle, zero = parabola, positive = hyperbola
• Converting to standard form requires completing the square; group $x$ terms and $y$ terms separately

Eccentricity Formula

• Formula: $e = \frac{c}{a}$—measures how "stretched" a conic is from circular
• Classification by eccentricity—$e = 0$ (circle), $0 < e < 1$ (ellipse), $e = 1$ (parabola), $e > 1$ (hyperbola)
• Higher eccentricity = more elongated; Earth's orbit has $e \approx 0.017$, nearly circular

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Focus-Directrix Relationships

The directrix provides an alternative way to define parabolas using the distance ratio property.

Directrix for Vertical Parabola

• Equation: $y = k - p$—a horizontal line located $p$ units opposite the focus from the vertex
• Defines the parabola geometrically—every point is equidistant from focus $(h, k + p)$ and this directrix
• Perpendicular to axis of symmetry; the directrix and focus are always on opposite sides of the vertex

Directrix for Horizontal Parabola

• Equation: $x = h - p$—a vertical line located $p$ units opposite the focus from the vertex
• Same distance property applies—point-to-focus distance equals point-to-directrix distance
• Sign of $p$ matters—if $p > 0$, focus is right of vertex and directrix is left; flip for $p < 0$

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Quick Reference Table

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Self-Check Questions

1. Given the equation $\frac{(x-2)^2}{16} + \frac{(y+3)^2}{25} = 1$, is the major axis horizontal or vertical? What is the center?

2. How can you tell the difference between an ellipse and a hyperbola just by looking at the standard form equation?

3. Compare and contrast the formulas for $c$ in ellipses versus hyperbolas. Why does this difference make geometric sense?

4. A parabola has vertex at $(1, -2)$ and focus at $(1, 0)$. Write its equation and find the directrix.

5. You're given $4x^2 + 9y^2 - 16x + 18y - 11 = 0$. What type of conic is this, and what's your first step to convert it to standard form?